Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for the Compressible Navier-Stokes Equations

Due to the lack of sufficiently strong a priori estimates to yield global-in-time smooth solutions, the compressible Navier-Stokes system suffers the same deficiency as most of its counterparts in continuum mechanics. On the one hand, in most cases for initial boundary value problems of multidimensional compressible Navier-Stokes equations with initial data of arbitrary size, only local well-posedness is known. On the other hand, the theory of global existence of weak solutions has been established by P. L. Lions in the barotropic case and developed by E. Feireisl and A. Novotný for the full system. The present chapter concerns with two interrelated aspects of compressible Navier-Stokes equations: blow-up mechanism of local strong solutions and conditional regularity of global weak solutions. For the barotropic Navier-Stokes equations, it is shown that the upper bound of the density must blow up provided the local strong solution breaks down at some time. For the full Navier-Stokes-Fourier system of ideal fluids, the regularity of local strong solutions is controlled by the upper and lower bounds of the density as well as the bound of the temperature. While for Navier-Stokes-Fourier system with more general structure, it is shown that Lipschitz bound of the velocity controls the regularity of strong solutions, which is similar to the classical situation on incompressible Navier-Stokes equations. Combining with the property of weak-strong uniqueness to weak solutions, it follows that the same conditions guarantee the regularity of weak solutions.